Atlas › Modular forms › Weight 2 › Level 333 › 333.2.a

Newspace 333.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(333))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
333.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	+	333c1 rank 1
333.2.a.b	1	$\mathbb{Q}$	$q - 2q^{4} - q^{7} + O(q^{8})$	−	333a1 rank 1
333.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} - 4q^{7} + O(q^{8})$	+	333b1 rank 1
333.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 2q^{5} - q^{7} + O(q^{8})$	+	333d1 rank 0
333.2.a.e	3	degree 3	$3q - 3q^{2} + 5q^{4} - 4q^{5} - 4q^{7} + O(q^{8})$	−
333.2.a.f	4	degree 4	$4q + 4q^{4} + 8q^{7} + O(q^{8})$	−
333.2.a.g	4	degree 4	$4q + 4q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13} + 11T_{17}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
333.2.a.a	$x - 5$
333.2.a.b	$x + 128$
333.2.a.c	$x + 81$
333.2.a.d	$x - 20$
333.2.a.e	$x^{3} + 125x^{2} + 655x - 232403$
333.2.a.f	$x^{4} - 112x^{3} + 3930x^{2} - 44464x + 83683$
333.2.a.g	$x^{4} - 84x^{3} - 9842x^{2} + 484630x + 1061149$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(333))$	15
Old forms	20
Cusp forms $S_{2}(\Gamma_0(333))$	35
Eisenstein series $E_{2}(\Gamma_0(333))$	7
Modular forms $M_{2}(\Gamma_0(333))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(333)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 9, 37, 111, 333]: [0, 1, 2, 3, 14, 40] (certified at ell = 2147483629)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (13, 9), (17, 11)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(333, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 333:2 1000

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